Adiabatic potential energy curves of the 31Σ+, 33Σ+, 21Π and 23Π states correlating for large internuclear distance with the K(4s) + Li(2p) atomic asymptote were calculated. Very good agreement between the calculated and the experimental curve of the 21Π state allowed for a reliable description of the dissociation process through a small (∼20 cm-1 for J = 0) potential energy barrier. The barrier supports several rovibrational quasi-bound states and explicit time evolution of these states via the time-dependent nuclear Schrödinger equation, showed that the state populations decay exponentially in time. We were able to precisely describe the time-dependent dissociation process of several rovibrational levels and found that our calculated spectrum match very well with the assigned experimental spectrum. Moreover, our approach is able to predict the positions of previously unassigned lines, particularly in the case of their low intensity.